The main body of the first edition of this paper is published in Games and
Economic Behavior, 28 (1999), 155-170 . Copyright and all rights
on the published paper therein are retained by Academic Press (Copyright
1999 by Academic
Press). Copyright and all rights on the parts
of this document which are not published in the paper therein are retained
by the author (Copyright 1999 by Ariel
Rubinstein).

Abstract

This is a revised version of my
paper with the same title published in Games and Economic Behavior,
28 (1999), 155-170. The paper summarizes my experience in teaching an
undergraduate course in game theory in 1998 and in 1999. Students were required
to submit two types of problem sets:pre-class problem sets, which served
as experiments, and post-class problem sets, which require the students to
study and apply the solution concepts taught in the course. The sharp distinction
between the two types of problem sets emphasizes the limited relevance of
game theory as a tool for making predictions and giving advice. The paper
summarizes the results of 43 experiments which were conducted during the
course. It is argued that the crude experimental methods produced results
which are not substantially different from those obtained at much higher
cost using stricter experimental methods.

My sincere
thanks to my two excellent research assistants, Yoram Hamo, during the 1998
course, and Michael Ornstein, during the 1999 course and in the writing of
the paper and its revised version.

1.
Introduction

Teaching game theory to undergraduates
has become standard in economics and other social science disciplines . This
is "great news" for game theorists. Academic knowledge is created and circulates
within a small circle of researchers for a very long time before the "happy"
moment it enters undergraduate textbooks. In the case of game theory, that
moment should not only be a cause for celebration. As game theorists, we
have a responsibility for the way it is taught. We are the only people who
can control and influence the content of the material taught. In particular,
we have a duty to control the message that game theory transmits to the broader
community.

It is my impression that most students
approach a course in game theory with the belief that game theory is about
the way that game situations are played and that its goal is to predict strategic
behavior. They hope the course will provide them with the tools to play game-like
situations better. During the course, they are disappointed with the poor
performance of game theory on the descriptive level and its lack of relevance
to practical problems.

The disappointment, of course,
leads the student to ask "what is the relation between the 'game theoretic
prediction' and the real world?" My impression is that undergraduate textbooks
are vague on this question. This is probably because we, game theorists,
are confused about what the theory is trying to accomplish.

In the past, I held the "radical"
view that undergraduate studies of game theory may influence students
negatively. Students may recognize the legitimacy of manipulative
considerations. They may come to believe that they need to use mixed
strategies. They may tend to become more suspicious and to put less
trust in verbal statements . They may adopt game theoretic solutions
dogmatically. However, a pilot experiment which I conducted together with
a group of graduate students at Tel Aviv University (Gilad Aharanovitz, Kfir
Eliaz, Yoram Hamo, Michael Ornstein, Rani Spiegler and Ehud Yampuler) made
me less certain about this position. When we compared the responses of economics
students to daily strategic situations before and after a course in game
theory, we found little difference before and after the course, though there
was a clear correlation between their responses and their second
major.

My method of teaching an undergraduate
course in game theory is derived from my views on the relationship between
theory and real life. I perceive game theory as the study of a set of
considerations used (or to be used) by people in strategic situations. I
do not, however see our models as being in any way constructions or depictions
of how individuals actually play game-like situations and I have never understood
how an equilibrium analysis can be used as the basis for a recommendation
on how to play real "games". My goal as a teacher is to deliver a loud and
clear message of separation between game theoretic models and predictions
of strategic behavior in real life.

Students in my class were asked
to complete two types of assignments. The "post-class" problem sets were
standard exercises that cane be found in any game theory text. Students were
asked to fit games to verbal situations, apply standard game theoretic solution
concepts and investigate them analytically. The answers to these problem
sets were categorized as right/wrong.

The new feature here was that students
were also requested to respond regularly to "pre-class" problems sets, which
were posted weekly in the course
website. The responses were collected in a log file, which allowed me
to enter the class with statistics regarding the results. It was stressed
that there are no right/wrong answers to the pre-class problems. The results
were compared with the standard game theoretic treatment, and it was pointed
out that some of the results fit the game theory analysis well while many
others do not. Game theory was described as a collection of considerations
which could or could not be used.

The pre-class problem sets served
two additional purposes: First, they helped the students to concentrate on
the examples discussed later in class. It is not a trivial task for students
to absorb several games in one class and this method facilitated their
understanding of the material taught. Second, they provided a cheap and
convenient tool for experimentation. I am fully aware of the potential for
criticism of this method: No monetary rewards were offered. However, comparisons
between the results achieved in class with those received in more standard
frameworks show, in my opinion (and I know this may be controversial),
insignificant differences.

Whereas the first version of this
paper included the results in the 1998 course only, I am including the results
of both 1998 and 1999 in the new version. Most of the experiments in 1999
repeated those in 1998. Very few experiments are new. Several versions were
changed, sometimes in order to test some framing effect. In 80% of the
experiments which were repeated in both years, the results were remarkably
similar. In only two experiments,
"he
will play first" and
"
randomization2") are the results qualitatively different. In the first
case, the result last year was suspicious and I do not rule out a technical
mistake in processing the information. I do not have an explanation for the
second case. In any case, I find the comparison of the results in the two
classes important as a self-disciplinary devise to detect such mistakes.

The following table summarizes
almost all the experiments conducted (though not in the order presented).
I omitted only those 1998 experiments which contained a clear problem in
their wording. The reader may view the experiment by clicking the box in
the left-hand column. The experiment page is linked to a result page. The
second column is anchored to a discussion of the experiment in this
text.

Note:you need javascript to be enabled in order to see this paper
properly.

The games in this category were
meant to introduce the students to basic strategic considerations. The students'
attention was directed to considerations which affect the outcome of a game
but are excluded from the game theoretic analysis.

The game guess
the 2/3, where a subject must announce a number between 0 and 99 with
the aim of guessing "the highest integer which is no higher than 2/3 of the
average of all the responses", has become a standard tool for demonstrating
game theoretic considerations ("I think that they think that...") and pointing
out the tension between real-life behavior and analysis. The game has a unique
Nash equilibrium outcome in which all players choose the number 1. The
results
fit well with those in the literature (see for example,
Camerer (1997), Nagel
(1995) and Thaler (1998)). The winning number
was 19 in 1998 and 23 in 1999. (Nagel's winning number was higher at 24,
and Thaler's large experiment with Financial Times readers received
a result of 13). Our result is lower than Nagel's due to the fact that a
significant number of students (versus almost no one in Nagel's experiment)
chose the lowest number. In Thaler's experiment, I suspect that the relatively
low number is an outcome of the participants' bias in favor of "more
sophisticated" subjects.

The full game theoretical analysis
of this particular game is not trivial and therefore, in 1999 I offered part
of the class a
modified
version of the question where the winning rule was changed to "guess
2/3 of the average of the other players". The
results
were not significantly different.

The next
game
was meant to test whether player 1 is certain that player 2 will take the
action which is "clearly" optimal for player 2.

A

B

A

5, 5

-100, 4

B

0, 1

0, 0

The outcome (A,A) is the best outcome
for both players, and player 2 has no reason to "punish" player 1 by playing
B. Nevertheless, player 1 may be uncertain as to whether player 2 will employ
the correct reasoning.
Results:
In both years less than a quarter of the students did not trust the other
player and chose the safe action B. Beard and Beil (1994)
tested a similar effect in a two-stage extensive game where player 1 could
either take a safe action or allow player 2 the option of making a choice.
If player 2 makes the irrational choice, player 1 suffers a loss. Though
the payoff numbers are different, the results here are in line with those
in experiment 6 in Beard and Beil (1994).

The question whether players follow a more complicated process of successive
elimination of dominated strategies was
tested in
a 4X4 matrix game. Though no student
chose
the weakly dominated action "A", only 34% of the subjects chose "B", the
only action which survives the successive elimination process. The
choice of C and D is probably a result of high payoffs attached to some entries
in the rows of those actions.

The battle of the sexes was used to explore several fundamental
issues.

A

B

A

2, 1

0, 0

B

0, 0

1, 2

In 1999 students
were asked
to play the battle of the sexes where the row player was called "He" and
the column player was called "She". Students were asked to play the game
in the role of the player who fits their gender.
Results:
68% of the students chose their preferred action. Since we asked students
to play the game in the role which fits their real gender, we can compare
the choice of the students according to their gender. 75% of the males chose
their preferred action whereas the females were divided equally between the
two actions. In comparison, in Cooper, DeJong, Forsythe
and Ross (1993), about 63% of the subjects in any of the two roles chose
their preferred action. I am not aware of any study of the battle of the
sexes where the report of the results classified the subjects by their
gender.Back to the table

An essential assumption of simultaneous games is that each player makes a
move independently of the other. Thus, information that one of the players
plays at an earlier point of time is not a part of the model as long as the
other player plays the game without being informed about the taken already
action. Is this a significant piece of information? In the
next
game, a student plays the role of player 2 in a battle of the sexes
game:

He is told that player 1 moves
first, without informing him of player 2's action.
Cooper, DeJong, Forsythe and Ross (1993) found
(with slightly different numbers) that 70% of the players in the role of
player 2 chose A whereas in a standard simultaneous game, the proportion
was only 35%. (Note that in their experiment, each player played the game
a large number of times).
Here,
in 1999, I got the same results as those of Cooper,
DeJong, Forsythe and Ross (1993) whereas in 1998 I received strangely
a very different result (was it just a mistake of mine?).

Information about outcomes of similar situations played in the past
is another example of an additional type of information which is not included
in the description of a game. In
the next
game, a situation which fits the battle of the sexes was described verbally.
Students were asked to imagine that in the last play of a similar game, they
had conceded, in other words, chosen the "inferior" action. This piece of
information
made
the subjects (especially in 1999) choose their preferred action in proportions
higher than that they did in the standard battle of the sexes.

The next four problems tested signaling
effects in the battle of the sexes.
In cheap_talk1
, player 2 has just announced that he will play his favorite action. This
statement
was
sufficient for almost all students (80% in 1998 and 94% in 1999) who
were posted in the role of player 1 to believe the announcement and to play
B. In comparison, the effect was even stronger in
Cooper, DeJong, Forsythe and Ross (1980) where
96% of the subjects played B. (However, note again that the results there
are reported for the case where each player played the game a large number
of times).

In cheap_talk2,
a conversation takes place in which player 1 announces that he will play
his favorite action (T), player 2 insists that he will play his favorite
action (R), and player 1 responds by remaining silent. Is silence interpreted
as "agreeing" or as "disagreeing"? In this case, students were
asked to predict the outcome of the game. The
results
of the experiment ,conducted only in 1998 : A vast majority (85%) of the
subjects predicted the outcome (T,L), implying that they interpreted silence
as a confirmation of player 1's first announcement. The closest comparable
game is found in Cooper, DeJong, Forsythe and Ross
(1980), where players made their moves after a cheap talk stage in which
they made simultaneous announcements. In those cases where one player announced
T and the other remained silent, 80% of the outcomes were indeed
(T,L).

In cheap_talk3
(conducted only in 1999), I tried to investigate the way that subjects interpret
"silence". Subjects were asked to predict the outcome of the battle of the
sexes after player 2 had an opportunity to make an announcement and remain
silent. My guess is that "silence"
was
interpreted as "weakness". I think that the subject investigated is of much
interest; however, no definitive conclusions could be drawn from this single
experiment. The least which is needed to draw any meaningful conclusions
is to compare the results with an experiment where the subject is asked to
predict the outcome of the standard battle of the sexes but where the players
are named "1" and "2".

Ben-Porath and Dekel (1992) provide the setting for
the next
problem.
Player 1 is notified that player 2 did not burn money although he could have
done so. I doubt if any of the students had in mind the considerations which
Ben-Porath and Dekel described; however, the
results
were in the "right" direction moderately: 46% of the students in 1998 and
33% in 1999 chose action B, only a slightly more than expected without this
information. An interesting comparison with the results can be made to the
problem studied in Cooper, DeJong, Forsythe and Ross
(1980), where the mere existence of an outside option for player 2
(independent of the values of that option) resulted in a stronger tendency
for player 1 to "yield" than in the standard BoS.

The
next problem
in this group is quite different: I attempted to test the students' intuitions
as to whether eliminating a player's action can be harmful for that player.
In
1999 71% of the students expressed an intuition that the removal may
help, whereas in 1998, (with a more confused statement of the problem), the
group was split equally in their answers. I incorporate the question here
primarily to emphasize the point that pedagogically, it is interesting to
survey intuitions before discussing them formally in class.

Coordination games are well suited to experiments. The common finding is
that people succeed in coordinating on the salient option (see for example,
Mehta, Starmer and Sudgen (1994)). The question now
is what are the characteristics of the salient option.
Here,
subjects were asked to coordinate on one of four alternatives labeled Fiat
97, Fiat 96, Saab 95 and Fiat 97. I wanted to test a conjecture made by Michael
Bacharach: When each alternative is described in terms of a number of
characteristics, the salient option is the one which is distinctive in most
of the characteristics. My own conjecture was that the salient option is
the one which is most distinctive from the set of most common alternatives
(the Fiats in this experiment). This question was presented only in 1998
and
the
findings are that Michael was right!

In 1999 I tried
another
problem which demonstrates that it is not so easy to coordinate in cases
where each option is described by a vector of characteristics. In the experiment,
each subject has to choose one of the six alternatives (Gad,4,c), (Gad,3,a),
(Dan,4,a), (Gad,4,d), (Dan,3,e) and (Gad,4,b). The alternatives were presented
in a vertical list. The order of the top and bottom options was exchanged
for half of the class. According to the
results,
if two subjects were randomly chosen to play the game, the chances that they
would succeed to coordinate was 33% in the first order and 20% in the second
order, not very high, although higher than the 17% expected if the subjects
were choosing their actions uniformly randomly. In the results, we can observe
a strong tendency to avoid both the bottom option and the alternative (Gad,4,d)
which is received as the most undistinctive alternative.

The class of zero-sum games is
attractive as a teaching device since students are familiar with such games
from daily life. Given the sharply defined predictions (in payoff terms)
of equilibrium in zero-sum games, comparing results with equilibrium is simple.
All the problem sets were given to the students prior to the classroom discussion
of the notion of mixed strategies.

Immediately after responding to this problem, the students were asked to
play the game in the role of the
column
player.
In
both years , 86% of the students chose the action R, well above the
"predicted" 67%. I suspect that the fact that player 2s payoffs were
presented in negative numbers was the main reason for this finding. The results
inspired several interesting explanations. One of the more interesting stated
that in the second game, many subjects sought to justify their previous
choice.

A similar experiment is found in
Fox (1972), though it did not use negative numbers. Fox
found that the row player's distribution is close to 50-50 whereas the column
player is concentrated on R.

In the next
game (presented only in 1998), a subject chooses a number in the
interval [0,100] while aiming to be as close as possible to his opponent,
who wants to avoid him. The game has many equilibria. Every choice
is consistent with some equilibrium. A clear majority of the students
chose
the middle or the edge points.

Prior to the presentation of the maxmin theorem, students
were asked
to express their views as to whether "minmax is greater than maxmin" or "maxmin
is greater than minmax". The maxmin criterion was presented as a pattern
of reasoning whereby a player thinks that his opponent will always successfully
predict his action. The minmax criterion was presented as a pattern of reasoning
where the subject is a "magician" who always correctly guesses the other
players intentions, and the other player knows it. Though the minmax
is never below the maxmin, in both years, the
respondents
split almost evenly in their voting. This split demonstrates how difficult
and unintuitive this elementary inequality is.

The next two problems were designed to demonstrate systematic deviations
from the game theoretic predictions about zero-sum games resulting from
framing effects.
4_boxes
is a repetition of an experiment conducted by Rubinstein,
Tversky and Heller (1996). The subjects were asked to hide a treasure
in one of four boxes placed in a row and labeled A,B,A,A. The seeker is able
to open only one box. In 1998 the subjects were assigned the role of the
seeker. The distribution of
answers
(16%, 19%, 54%, 11%) is strongly biased towards the central A box, avoiding
the edges. These results were even more pronounced than those of the original
experiment (13%, 31%, 45%, 11%). In 1999 the subjects were assigned the role
of the hider. Once again, the results (16%, 18%, 45%, 22%) were similar to
the results (9%, 36%, 40%, 15%) obtained originally. Where a random seeker
of the 98's class was playing the game against a random player of the 99's
class the chances that he would find the prize was 33%, much above the game
theoretic "prediction" of 25%. By the way, the presentation of the results
in class created a feeling of real "discovery".

The strong tendency to avoid the edges was also obtained
in "hide a
treasure in a 5X5 table", where the subject hides a treasure in one of
the table's 25 boxes. Here, the 64% of the boxes placed at the
edges received only 47%-49% of the choices. This is certainly in line with
the results of Ayton and Falk (1995), who asked subjects
to hide three treasures in such a layout. They found avoidance of
the edges and strong concentration on boxes B4 and D4 (in our case, the most
frequent choices were D2 and D4).

The
dictator
game
illustrates
two principal modes of behavior: "taking the entire sum of money" (52%) and
"sharing it equally" (35%). In other words, half of the class (in 1998)
exhibit preferences which are not purely monetary in nature. The concentration
of subjects in the above two modes of behavior is similar to the results
of Forsythe, Horowitz, Savin, and Sefton (1994).
They found that even when the players played for real money, only 35% chose
to grab the whole sum. In
1999, the
question was framed slightly differently: a subject had to choose the
sum of money he gives to the other player (rather than the sum he takes for
himself). I did not find any significant difference. Still, 37%
chose
to split the sum equally and a bit more than half the class "grabbed" the
entire sum.

I think that no other game has been used in more experiments than the
ultimatum
game. Having to agree on the partition of 100 shekels, the offers in 1998
were
split into three groups: About 35% of the offers equalled 1, 39% offered
50, and 26% offered a sum between 10 and 40. In comparison, previous experiments
such as those of Guth, Schmittberger and Schwarze
(1982) and Forsythe, Horowitz, Savin, and Sefton
(1994), found that with or without the payment of real money, a higher
percentage of subjects offered an equal split of the pie and almost no subjects
offered to retain almost all of the money. In
1999,
the game was framed differently: the proposer had to declare the amount of
money he demands to himself.
Once again
a framing effect was not traced. About 1/3 of the subjects demand half the
sum and 39% (almost) demanded the entire sum.

Finally, when students were asked to determine a
minimal cutoff
point for acceptance when facing an unknown offer, about one half of
the the students in both years
said
that they would accept all offers (other than 0) and 35% of the students
in 1998 and 23% in 1999 set the cutoff point at half the divided sum. Previous
results ( see Harrison and McCabe (1992)) showed
a much higher proportion of subjects setting the cutoff point at
50%.

Although there is no difference
in the basic modes of behavior appearing among the students and the subjects
in laboratory experiments, it seems that the students in my class were more
aggressive than the subjects in the previous experiments. This is not
a very surprising fact, considering the prevailing mood in Israel (see
Roth, Prasnikar, Okuno-Fujiwara, and Zamir
(1991)).

In the first problem in this category, subjects were presented with a situation,
described verbally, similar to the one-shot chain store paradox game: A tailor
is
considering transforming
his shop into a mini-market in a location where a grocery store already
exists. He is afraid of a possible harmful response from the grocery store,
one which will be harmful to the grocery store as well.
About
53% of the students in 1998 and 58% in 1999 recommended that the tailor "enter"
the food market (in line with the sub-game perfect equilibrium). In contrast,
Schotter, Weigelt and Wilson (1994) found that a
much higher percentage of subjects chose to enter. The difference, in my
opinion, is due to the fact that the problem here was presented verbally
whereas Schotter, Weigelt and Wilson (1994) presented the subjects with an
explicit tree which assisted them in the backward inductive reasoning. In
fact, when they gave the subjects the opportunity to play the corresponding
normal-form game, only 57% of the subjects chose to enter.

The next problem was intended to demonstrate that in a game situation, more
rather than less information may be harmful. Students
were asked
how much they were willing to "pay" to exchange a play of the battle of the
sexes game for a play in a similar game in which the subject would be informed
"publicly" about the other player's move before the subject made his own
choice. The three equilibria payoffs for the BoS version are 20, 10
(pure equilibria payoffs) and 6.7 (a mixed equilibrium payoff), which is
also its maxmin value. The value of the only sub-game perfect equilibrium
of the extensive game which fits the alternative game is 10. Thus, even under
the most pessimistic view, a player should not value the offer at more than
3.3. Yet, 56% of the students in 1998 (and 51% in 1999)
were
ready to pay more than 3.3 and only 26% of the students in 1998 (and
42% in 1999), found the offer valueless.

In the next
problem I followed the ideas of Camerer, Johnson,
Rymon and Sen (1993), who performed one of the most beautiful experiments
I have ever come across. Subjects had to choose the order by which they would
expose information in a two-stage extensive game. The payoff numbers were
chosen to be complicated (some were negative and had many digits after the
decimal point) in order to create the impression that analyzing the game
was not a trivial task and required memory. Revealing "B" first
makes the analysis easier. However, in both years, only 36% of the subjects
analyzed the game from it's end, whereas 64% first investigated the content
of consequence A. This is definitely in line with the conclusion of Camerer,
Johnson, Rymon and Sen (1993) that people tend to analyze an extensive game
forward rather than backward.

Discounting: Students were asked to predict the outcome of a bargaining
session between two bargainers possessing identical characteristics except
that one is more impatient than the other.
Results:
about half of the students in each year, predicted an equal split. The other
subjects in 1998 were equally divided as to whether the more impatient person
would get more or less than half of the sum whereas in 1999 there was a tendency
to predict that the more impatient bargainer will get more than the patient
bargainer. Thus, the results do not confirm the intuition that people evaluate
impatience as a negative factor in bargaining. This result is in line with
the results of Ochs and Roth (1989), who demonstrated
the negligible effect of different discount rates on bargaining
outcomes.

Being
a proposer or being a responder: Game theoretic models suggest that being
a proposer provides a strategic advantage over being a responder. However,
students
seem
to consider the role of the responder more attractive. The contrast between
this finding and the standard game theoretic models begs for an
explanation!

Reputation:
A seller has refused several offers made by the subject. How does the subject
interpret the rejections?
Results:
62% of the students in 1998 (and 49% in 1999) did not find the refusal
informative, but almost all the rest considered the refusal to be an indication
that the value of the item is higher than what was initially
thought.

The first game in this class was the 100-period
centipede
game.
The results: Very few students (about 10%) chose
to stop the game immediately, as Nash equilibrium "predicts", 50% of the
students in 1998 and 60% in 1999 chose to "never stop" and 22% chose to stop
the game at the last or the penultimate opportunity. The closest comparison
to these results is Nagel and Tang (1998), who found
(in a six-period centipede game) that almost no subjects were following Nash
equilibrium strategies and that the vast majority of subjects were stopping
two or three periods before the end of the game. (See also McKelvey
and Palfrey (1992)).

Not all problems were intended to refute the game theoretic "predictions".
In a
game
with a similar structure to that of the centipede game, the subject was the
first in a sequence of players to decide whether "to stop" or "to pass the
game to the next player in line", with payoffs that made "all players pass"
the unique sub-game prefect equilibrium.
Results:
about 60% of the students indeed chose "pass".

In two other experiments, students
were asked to play a four-period repetition of the prisoner's dilemma game
and of the battle of the sexes. Students were asked to specify their strategies
as "plans of actions", that is, they were not asked to specify actions following
histories which contradict their own plans. The objective was to emphasize
the contrast between the formal concept of a strategy and the intuitive notion
of a strategy as a plan of action (see Rubinstein
(1991)).

In the repeated
PD, only 34% of the subjects in 1998 and 42% in 1999 chose C in the first
period. The
results
contain a large number of strategies, many of which were difficult to interpret.
This fact motivated me to ask the students in 1999 to describe their strategies
in words, as well. About half of the students in 1998 and 25% in 1999 chose
to play constant D , 5% of the students in each of the years chose to play
constant C, and 5% in 1998 and 13% in 1999 chose the Tit-for-Tat strategy.
The closest previous experiment is Selten and Stoecker
(1986); however, the results are difficult to compare.

In the repeated
BoS, only 10% of the subjects in 1998 (and 28% in 1999) started the game
by playing the less favorable action. Once again, the
results
contained a large variety of strategies. Two strategies were most frequent:
22% of the students started the game by playing the more favorable action
in 1998 (and 25% in 1999) and continuing to play the best response against
their opponent's last played action; 12% of the in 1998 (and 8% in 1999)
students chose the strategy "play the favorable action unless, in the past,
the opponent played his favorable action in a strict majority of the periods"
.

It is interesting to note that
in the last two experiments, although asked to describe their
strategies, many of the students chose to the describe their
considerations.

In class I presented various
interpretations of mixed strategies (see
Osborne
and Rubinstein (1994, ch. 3)): · the "naive" interpretation
- a player chooses a random device such as a roulette.· the "purification" idea
- a players behavior depends deterministically on unobserved
factors.· the "beliefs" interpretation-
a player's mixed strategy is what other players think about a players
behavior.· the "large population"
interpretation - a mixed strategy is a distribution of the modes of behavior
displayed by a large population of players who are matched randomly in order
to play the game.

Adopting either of the first two
interpretations led to a discussion of "random behavior". My aim was to
demonstrate to the students that when people choose "random behavior", they
create patterns of behavior which are not so random.

In
randomization
1, students were asked to choose an integer between 1 and 9. In
Simon (1971) and in Simon and Primavera
(1972), the number 7 was clearly the most frequent choice (33% and 24%
of the subjects, respectively, chose 7 from among the numbers 0,1,...,9).
Here, the conjecture that people over-choose "7" was
confirmed.
In fact, "7" and "5" were the most frequent choices at 17% each in both years.
Another clear phenomenon is the avoidance of the "edges" ("1" and "9"), which
were the least chosen alternatives.

In randomization2,
students were given the following task: "Randomly choose 4 of the integers
{1,2,...,8}." In 1998, three patterns were observed in the
results:

1) Out of the 70 possible answers,
only two were chosen by more than 3 students: 7 students chose "1234" and
6 chose "1357".2) Although 21% of the possible
sequences do not include either 1 or 2, only 6% of the students actually
excluded either 1 or 2 from their chosen sequence.3) The number 6 was chosen by 30%
of the students, far less than all other numbers. Even when we exclude the
students who chose the sequences "1234" or "1357", the proportion of answers
which did not include 6 was well above 50%. One explanation is that subjects
who chose three numbers from 1,2,3,4,5 felt they must also choose one of
the last two numbers: thus, they skipped 6. In 1999, the results were entirely
different. No student chose the sequence "1234" and only 2 students chose
"1357". One common observation: the number 8 was chosen well below 50%. This
is one of the few experiments I conducted in class in which I cannot explain
the sharp differences between the results in the two years.

Excess randomization is also well documented in the literature, under the
name "matching probabilities".
In "guess
the departments" (following an idea I was working on with the late Amos
Tversky), students had to guess the second major of five randomly chosen
students who study economics in a double-major program. Though the distribution
of the second major was not given to the students, one could expect that
they had some idea, and, in any case, maximization of the probabilities to
win the prize should have led them to choose the major they believe is the
most frequent among their five guesses.
Theresults clearly demonstrate an
excess use of randomization. Only 28% of the students in 1998 and 6% of the
students in 1999 repeated the same guess five times. All the others diversified
their answers and included at least 3 different choices in their list of
guesses. The results correspond well with Loewenstein
and Read (1995), who demonstrated a strong tendency towards diversification
when subjects had to choose a sequence of three items even though one item
was viewed by them as superior to the others.

In the next experiment, students were asked to choose one out of four
possibilities, which yield prizes with probabilities of 21%, 27%, 32% and
20%, respectively. The two versions presented in the two years were quite
different.
In 1998,
the students had to choose at which gate to choose a friend and
in 1999,
at which gate to ambush a suspect. In both years the students were explicitly
allowed to randomize but in 1998, one of the explicit options was to choose
one of the gates. The differences in the responses to the two versions is
very clear.
In 1998,
only 28% of the students chose to randomize.
In 1999,
once the students had explicitly to allocate the probabilities among the
four options, 67% chose to randomize (in particular, 30% chose the numbers
which were given as the probabilities of each gate and 14% assigned equal
probabilities to the four gates).

The problem
of count
the number of F's was distributed this year on the Net (I do not know
who initiated this beautiful problem). A subject was asked to count the number
of F's in a 90-letter text. The question was given, first of
all,...for but it was also
supposed to remind the students that people often make systematic mistakes.
Counting the number of "F's" in an 81- letter text is supposed to be a trivial
task, but
only one third of the students did it right. The common answer (at least
among my friends), "3", was given by a quarter of the students in 1998 and
30% in 1999.

The next problem is the simplest problem yet devised which demonstrates the
winner's
curse phenomenon. A student has to bid for an object; he will receive
only it if he offers more than its real value, a number which is distributed
uniformly in the interval [0,1000]. The bidder will then be able to sell
the item for 150% of its real value. The problem was studied first (with
and without real money) by Samuelson and Bazerman
(1985). The
results
here were not significantly different (the differences seem to depend on
the fact that I allowed the subjects to bid for more than 1000). Only about
10% of the students gave the "optimal" offer 0; the majority of the students
in each of the two years offered 500 or more!

Another puzzle which has been widely discussed in the last few years, is
the exchange of
envelopes.
Two positive numbers, one twice the size of the other, are placed in two
sealed envelopes. The subject randomly receives one of the envelopes and
another person tries to persuade him to exchange the envelopes with the
claim: "If the number x is in your envelope, the expected value of
the other envelope is 3x/2". This problem was offered to the students
at a late stage in the course followed by a post-class problem where the
students had to understand the Brams and Kilgour
(1995) game theoretic treatment of the problem. In each of the
two years, only about 20% of the students
expressed,
willingness to exchange envelopes.

Some other problems dealt with
the basic assumptions of the VNM theory of decision making under
uncertainty.

A variant
of the Allais paradox (originating in Kahneman and Tversky
(1979)), was presented to the students. Students had to choose between
two lotteries: one which yields $4,000 with probability 0.2 and a second
which yields $3,000 with probability 0.25.
The results: 72%-74% of the students chose the
first lottery, very close to the results of Kahneman and Tversky (1979),
where 65% of the subjects chose the first lottery. In 1999, I confronted
the students with
the variant
choice between the certain $3,000 and a lottery which yields $4,000 with
probability 0.8.
The results : 86% of the subjects chose the
certain prize. As is well known, the two results sharply conflict with expected
utility theory.

A basic principle of rationality in decision making under uncertainty is
the "sure thing" principle: If action D is better than action C under any
of two exclusive circumstances, then D is better than C when the decision
maker does not have information what circumstance had occured.
Shafir and Tversky (1992) showed that more people choose
to cooperate in the prisoner's dilemma than in either of the two cases in
which they are told that the other player had cooperated or defected. Here,
the problems were given to the students in the
order "player
2 has made up his mind to cooperate"
, "player
2 has made up his mind to defect" and
a regular
prisoner's dilemma. The
results
were in line with Shafir and Tversky (1992): In 1998, only 9% cooperated
when the other player did so, only 4% cooperated when the other player defected,
and 16% cooperated when they did not know what the other player chose (12%,
0%, 16% were the corresponding numbers in 1999 and 3%, 16% and 37% were the
corresponding figures in Shafir and Tversky (1992), with similar though not
identical payoffs).

Does game theory affect the ethical
attitudes of students concerning behavior in strategic situations? The suspicion
may arise that game theory intensifies "selfish motivations", strengthens
manipulative attitudes, reduces the importance of ethical considerations,
and so forth. In collaboration with a group of students, Gilad Aharanovitz,
Kfir Eliaz, Yoram Hamo, Michael Ornstein, Rani Spiegler, and Ehud Yampuler,
we gave a series of questions to the students at the first meeting of the
course. We compared the results with those of a similar group of students
who had just completed a similar course given by another teacher. The results
did not show any clear difference between students' behavior before and after
taking the course. We still feel that more experiments should be conducted
on the subject. In the meantime, let it suffice to present the results of
two of our problems presented ti the students.

In order to examine the tendency of students to behave manipulatively in
elections, they were
presented
with some hypothetical election situations in which their candidates were
doomed to lose but where they could increase the chances of their second
best choice. Indeed, 76-79% of the students
were
prepared not to vote for their favorite candidate in order to help their
second-best choice to win. (For a related experiment see Eckel
and Holt (1989).)

In another problem, students were placed in the role of
auto
dealers who had offered a price and then received information that the
potential buyer was ready to pay more than they had offered him. They then
had to decide whether or not to raise their price.
Here,
55% of the students in 1998 and an even higher proportion, 71% in 1999, stated
that they would not raise the price.

The main purpose of this paper
was to summarize my experience in teaching an undergraduate course in game
theory. However, in retrospect, I feel that the experience presents an
opportunity to evaluate the experimental methods used in game theory. Researchers
are split into two camps: Some create careful laboratory environments and
pay the subjects monetary rewards for their performance in the experiment;
others ask subjects to fill out questionnaires requiring them to speculate
on hypothetical situations.

I fall into the second category.
(Compare with the view in Camerer and Hogarth
(1999)). My impression is that the results are as significant as
those obtained under more sterile conditions, in the sense that the same
modes of behavior appear in both sets of results. If we were interested in
obtaining precise statistics regarding the appearance of those modes of behavior
in the general population, then both methods are deeply flawed since our
subjects are never chosen from random representative samples. In cases where
the previous results differ quantitatively (as they do in the dictator and
ultimatum games, for example), the distribution of modes of behavior is clearly
affected by culture, education and personal characteristics; hence, there
is no reason to expect uniform results. I would therefore like to stress
my doubts as to the necessity of laboratory conditions and the use of real
money in experimental game theory.